Infima Preserving Mapping on Filters is Increasing
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Theorem
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a mapping.
For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$.
Then $f$ is increasing.
Proof
Let $x, y \in S$ such that:
- $x \preceq y$
- $\set x$ and $\set y$ admit infima in $\struct {S, \preceq}$
By Infimum of Upper Closure of Set:
- $\set x^\succeq$ and $\set y^\succeq$ admit infima in $\struct {S, \preceq}$
where $\set x^\succeq$ denotes the upper closure of $\set x$.
- $x^\succeq$ and $y^\succeq$ admit infima in $\struct {S, \preceq}$
By Upper Closure of Element is Filter:
- $x^\succeq$ and $y^\succeq$ are filter in $\struct {S, \preceq}$
By assumption and definition of mapping preserves the infimum on subset:
- $\map {f^\to} {x^\succeq}$ and $\map {f^\to} {y^\succeq}$ admit infima in $\struct {T, \precsim}$
and
- $\inf \set {\map {f^\to} {x^\succeq} } = \map f {\inf \set {x^\succeq} }$ and $\inf \set {\map {f^\to} {y^\succeq} } = \map f {\inf \set {y^\succeq} }$
By Infimum of Upper Closure of Element:
- $\inf \set {x^\succeq} = x$ and $\inf \set {y^\succeq} = y$
By Upper Closure is Decreasing:
- $y^\succeq \subseteq x^\succeq$
By Image of Subset under Mapping is Subset of Image:
- $\map {f^\to} {y^\succeq} \subseteq \map {f^\to} {x^\succeq}$
Thus by Infimum of Subset:
- $\map f x \precsim \map f y$
Thus by definition:
- $f$ is increasing.
$\blacksquare$
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL_0:69